Curve (math)

In mathematics , a curve (from the Latin curvus “bent, curved”) is a one-dimensional object . In contrast to a straight line , for example , a curve does not have to be straight, but can instead take any shape.

One-dimensional means informally that you can only move in one direction (or the opposite direction) on the curve. Whether the curve lies in the two-dimensional plane ("plane curve") or in a higher-dimensional space (see space curve ) is irrelevant in this conceptual context.

Depending on the sub-area of ​​mathematics, there are different specifications for this description.

Parametric representations

→ Main article : Path (mathematics)

A curve can be defined as the image of a path . A path is (in contrast to colloquial language) a continuous mapping from an interval into the considered space, e.g. B. in the Euclidean plane . ${\displaystyle \mathbb {R} ^{2}}$

cubic curve with a colon. t  → ( t ²  - 1,  t  ( t ²  - 1)) or y ²  =  x ² ( x  + 1)

Examples:

• The image

${\displaystyle {[0,2\pi [}\to \mathbb {R} ^{2},\quad t\mapsto (\cos t,\sin t)}$

describes the unit circle in the plane.

• The image

${\displaystyle \mathbb {R} \to \mathbb {R} ^{2},\quad t\mapsto {\big (}t^{2}-1,t(t^{2}-1){\big )}}$

describes a curve with a single colon at , corresponding to the parameter values and . ${\displaystyle (0,0)}$ ${\displaystyle t=1}$ ${\displaystyle t=-1}$

Occasionally, especially with historical names, no distinction is made between path and curve. So the interesting structure in the Hilbert curve is the way; the image of this path is the unit square, so it no longer has any fractal structure.

The parameter representation gives the curve a sense of direction in the direction of the growing parameter.

Equation representations

→ Main article : Implicit curve

A curve can also be described by one or more equations in the coordinates. Examples of this are again the pictures of the two curves given by the parameter representations above:

• the equation

${\displaystyle x^{2}+y^{2}=1}$

describes the unit circle in the plane.

• the equation

${\displaystyle y^{2}=x^{2}(x+1)}$

describes the curve specified above in the parametric representation with a colon.

If the equation is given by a polynomial , as here , the curve is called algebraic .

Function graph

→ Main article : Function graph

Function graphs are a special case of both of the above forms: The graph of a function

${\displaystyle f\colon D\to \mathbb {R} ,\quad x\mapsto f(x)}$

can either be a parametric representation

${\displaystyle D\to \mathbb {R} ^{2},\quad t\mapsto (t,f(t))}$

or as an equation

${\displaystyle \Gamma _{f}=\{(x,y)\in \mathbb {R} ^{2}\mid y=f(x)\}}$

can be specified.

Is in school mathematics of Kurvendiskussion spoken, one usually thinks only this special case.

Differentiable curves, curvature

Let be an interval and a regular curve , i.e. H. for everyone . The length of the curve is ${\displaystyle [a,b]\subset \mathbb {R} }$ ${\displaystyle c\colon [a,b]\to \mathbb {R} ^{n}}$ ${\displaystyle |c'(x)|\neq 0}$ ${\displaystyle x\in (a,b)}$

${\displaystyle l=\int _{a}^{b}|c'(t)|\,dt}$

The function

${\displaystyle x\mapsto \int _{a}^{x}|c'(t)|\,dt}$

is a diffeomorphism , and the concatenation of with the inverse diffeomorphism yields a new curve with for all . One says: is parameterized according to the arc length. ${\displaystyle [a,b]\to [0,l]}$ ${\displaystyle c}$ ${\displaystyle {\tilde {c}}\colon [0,l]\to \mathbb {R} ^{n}}$ ${\displaystyle |{\tilde {c}}'(x)|=1}$ ${\displaystyle x\in (0,l)}$ ${\displaystyle {\tilde {c}}}$

Let be an interval and a curve parameterized according to the arc length. The curvature of at the point is defined as . For plane curves, the curvature can also be given a sign : If the rotation is 90 °, then it is determined by . Positive curvature corresponds to left turns, negative right turns. ${\displaystyle [a,b]\subset \mathbb {R} }$ ${\displaystyle c\colon [a,b]\to \mathbb {R} ^{n}}$ ${\displaystyle c}$ ${\displaystyle s}$ ${\displaystyle \kappa (s)=|c''(s)|}$ ${\displaystyle J}$ ${\displaystyle \kappa (s)}$ ${\displaystyle c''(s)=\kappa (s)\cdot Jc'(s)}$

Closed curves

Be a flat curve. It is closed when , and simply closed if, in addition to is injective. The Jordanian curve theorem states that a simply closed curve divides the plane into a bounded and an unbounded part. If a closed curve is common to all , the curve can be assigned a number of revolutions which indicates how often the curve runs around the zero point. ${\displaystyle c\colon [0,1]\to \mathbb {R} ^{2}}$ ${\displaystyle c(0)=c(1)}$ ${\displaystyle c}$ ${\displaystyle [0,1)}$ ${\displaystyle c}$ ${\displaystyle c(t)\neq (0,0)}$ ${\displaystyle t\in [0,1]}$

Smooth closed curves can be assigned to another number, the tangent revolution number , which for a parameterized by arc Läge curve by ${\displaystyle c\colon [0,l]\to \mathbb {R} ^{2}}$

${\displaystyle {\frac {1}{2\pi }}\int _{0}^{l}\kappa (t)\,dt}$

given is. The circulation rate of Heinz Hopf says that a simple closed curve tangent winding number or has. ${\displaystyle 1}$ ${\displaystyle -1}$

Generally be a topological space . Instead of closed paths with , one also speaks of loops with a base point . Because the quotient space is homeomorphic to the unit circle , one identifies loops with continuous maps . Two loops with a base point are called homotopic if they can be continuously deformed into one another while maintaining the base point, i.e. H. if there is a continuous mapping with , for all and for all . The equivalence classes homotopic loops form a group , the fundamental group of . If , then the fundamental group is isomorphic to over the number of turns . ${\displaystyle X}$ ${\displaystyle c\colon [0,1]\to X}$ ${\displaystyle c(0)=c(1)}$ ${\displaystyle c(0)}$ ${\displaystyle [0,1]/\{0,1\}}$ ${\displaystyle S^{1}}$ ${\displaystyle S^{1}\to X}$ ${\displaystyle c_{1},c_{2}}$ ${\displaystyle x}$ ${\displaystyle H\colon [0,1]^{2}\to X}$ ${\displaystyle H(s,0)=c_{1}(s)}$ ${\displaystyle H(s,1)=c_{2}(s)}$ ${\displaystyle s}$ ${\displaystyle H(0,t)=H(1,t)=x}$ ${\displaystyle t}$ ${\displaystyle X}$ ${\displaystyle X=\mathbb {R} ^{2}-\{0\}}$ ${\displaystyle \mathbb {Z} }$

Space curves

Let be an interval and a curve parameterized according to the arc length. The following names are standard: ${\displaystyle [a,b]\subset \mathbb {R} }$ ${\displaystyle c\colon [a,b]\to \mathbb {R} ^{3}}$

${\displaystyle {\begin{aligned}t(s)&=c'(s)\\n(s)&={\frac {t'(s)}{|t'(s)|}}\\b(s)&=t(s)\times n(s)\end{aligned}}}$

(defined whenever ). is the tangential vector, the normal vector and the binormal vector, the triple is called the accompanying tripod . The curvature is defined by the twist . The Frenet formulas apply : ${\displaystyle t'(s)\neq 0}$ ${\displaystyle t(s)}$ ${\displaystyle n(s)}$ ${\displaystyle b(s)}$ ${\displaystyle (t,n,b)}$ ${\displaystyle \kappa (s)=|t'(s)|=|c''(s)|}$ ${\displaystyle \tau (s)}$ ${\displaystyle b'(s)=-\tau (s)n(s)}$

${\displaystyle {\begin{matrix}t'&=&&\kappa n\\n'&=&-\kappa t&+&\tau b\\b'&=&&-\tau n\end{matrix}}}$

The main theorem of local curve theory says that a curve can be reconstructed from curvature and winding: If smooth functions are with for all (the value 0 is therefore not allowed for), there is exactly one corresponding curve except for movements . ${\displaystyle \kappa ,\tau \colon [0,l]\to \mathbb {R} }$ ${\displaystyle \kappa (s)>0}$ ${\displaystyle s\in [0,l]}$ ${\displaystyle \kappa }$

The three vectors of two , or spanned planes through the curve point wear special name: ${\displaystyle t}$ ${\displaystyle n}$ ${\displaystyle b}$

• The Oskulations flat or osculating plane is by and spanned. ${\displaystyle t}$ ${\displaystyle n}$
• The normal plane is spanned by and . ${\displaystyle n}$ ${\displaystyle b}$
• The rectifying plane or extension plane is spanned by and . ${\displaystyle t}$ ${\displaystyle b}$

Curves as independent objects

Curves without a surrounding space are relatively uninteresting in differential geometry , because every one-dimensional manifold is diffeomorphic to a real straight line or to the unit circle . Properties such as the curvature of a curve cannot be determined intrinsically either. ${\displaystyle \mathbb {R} }$ ${\displaystyle S^{1}}$

In algebraic geometry and, related to it, in complex analysis , “curves” are generally understood to be one-dimensional complex manifolds , often referred to as Riemann surfaces . These curves are independent study objects, the most prominent example being the elliptical curves . See curve (algebraic geometry)

Historical

The first book of the elements of Euclid began with the definition “A point is that which has no parts. A curve is a length without a width. "

This definition can no longer be maintained today, because there are, for example, Peano curves , i.e. H. continuous surjective mappings that fill the entire level. On the other hand, it follows from Sard's lemma that every differentiable curve has an area of ​​zero, that is, in fact, as required by Euclid, has “no width”. ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} ^{2}}$

literature

• Ethan D. Bloch: A First Course in Geometric Topology and Differential Geometry. Birkhäuser, Boston 1997.
• Wilhelm Klingenberg: A Course in Differential Geometry. Springer, New York 1978.

Web links

Commons : Curves  - collection of images, videos and audio files

Wiktionary: curve  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ^ H. Neunzert, WG Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: With an introduction to vector and matrix calculations . A textbook and workbook. 2nd Edition. Springer, 2013, ISBN 978-3-642-97840-1 , 23.5 ( limited preview in the Google book search).
2. ↑ H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematik . 12th edition. Walter de Gruyter, 1994, ISBN 978-3-486-78544-9 ( limited preview in Google book search).
3. ^ W. Kühnel: Differentialgeometrie. Vieweg-Verlag, 1999, ISBN 978-3-8348-0411-2 , section 2.9.